This article is cited in 75 papers

The class of groups all of whose subgroups with lesser number of generators are free is generic

G. N. Arzhantseva, A. Yu. Ol'shanskii

M. V. Lomonosov Moscow State University

Abstract: It is shown that, in a certain statistical sense, in almost every group with $m$ generators and $n$ relations (with $m$ and $n$ chosen), any subgroup generated by less than $m$ elements (which need not belong to the system of generators of the whole group) is free. In particular, this solves Problem 11.75 from the Kourov Notebook. In the proof we introduce a new assumption on the defining relations stated in terms of finite marked groups.

UDC: 512.5

Received: 05.01.1995

DOI: 10.4213/mzm1744

 English version: Mathematical Notes, 1996, 59:4, 350–355

Bibliographic databases:

• 
• 
•